Theorem canthwdom 9542

Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9119, equivalent to canth 7366). (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
canthwdom	⊢ ¬ 𝒫 𝐴 ≼* 𝐴

Proof of Theorem canthwdom

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elpw 5328	. . . . 5 ⊢ ∅ ∈ 𝒫 𝐴
2		ne0i 4295	. . . . 5 ⊢ (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅)
3	1, 2	mp1i 14	. . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝒫 𝐴 ≠ ∅)
4		brwdomn0 9532	. . . 4 ⊢ (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴))
5	3, 4	syl 18	. . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴))
6	5	ibi 270	. 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)
7		relwdom 9529	. . . . 5 ⊢ Rel ≼*
8	7	brrelex2i 5720	. . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝐴 ∈ V)
9		foeq2 6791	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝑥))
10		pweq 4577	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
11		foeq3 6792	. . . . . . . 8 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
12	10, 11	syl 18	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
13	9, 12	bitrd 282	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
14	13	notbid 321	. . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑓:𝑥–onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴–onto→𝒫 𝐴))
15		vex 3459	. . . . . 6 ⊢ 𝑥 ∈ V
16	15	canth 7366	. . . . 5 ⊢ ¬ 𝑓:𝑥–onto→𝒫 𝑥
17	14, 16	vtoclg 3523	. . . 4 ⊢ (𝐴 ∈ V → ¬ 𝑓:𝐴–onto→𝒫 𝐴)
18	8, 17	syl 18	. . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ 𝑓:𝐴–onto→𝒫 𝐴)
19	18	nexdv 1966	. 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)
20	6, 19	pm2.65i 196	1 ⊢ ¬ 𝒫 𝐴 ≼* 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4287  𝒫 cpw 4563   class class class wbr 5110  –onto→wfo 6536   ≼^* cwdom 9527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-wdom 9528

This theorem is referenced by:  pwdjudom  10199